This module presents a type for storing numbers in the log-domain.
The main reason for doing this is to prevent underflow when
multiplying many small probabilities as is done in Hidden Markov
Models and other statistical models often used for natural
language processing. The log-domain also helps prevent overflow
when multiplying many large numbers. In rare cases it can speed
up numerical computation (since addition is faster than
multiplication, though logarithms are exceptionally slow), but
the primary goal is to improve accuracy of results. A secondary
goal has been to maximize efficiency since these computations
are frequently done within a O(n^3) loop.

The LogFloat of this module is restricted to non-negative
numbers for efficiency's sake, see the forthcoming
Data.Number.LogFloat.Signed for doing signed log-domain
calculations. (Or harass the maintainer to write it already.)

Exceptional numeric values

LogFloat data type

A LogFloat is just a Double with a special interpretation.
The logFloat function is presented instead of the constructor,
in order to ensure semantic conversion. At present the Show
instance will convert back to the normal-domain, and so will
underflow at that point. This behavior may change in the future.

Performing operations in the log-domain is cheap, prevents
underflow, and is otherwise very nice for dealing with miniscule
probabilities. However, crossing into and out of the log-domain
is expensive and should be avoided as much as possible. In
particular, if you're doing a series of multiplications as in
lp * logFloat q * logFloat r it's faster to do lp * logFloat
(q * r) if you're reasonably sure the normal-domain multiplication
won't underflow, because that way you enter the log-domain only
once, instead of twice.

Even more particularly, you should avoid addition whenever
possible. Addition is provided because it's necessary at times
and the proper implementation is not immediately transparent.
However, between two LogFloats addition requires crossing the
exp/log boundary twice; with a LogFloat and a regular number
it's three times since the regular number needs to enter the
log-domain first. This makes addition incredibly slow. Again,
if you can parenthesize to do plain operations first, do it!

Accurate versions of logarithm/exponentiation

Definition: log1p == log . (1+). The C language provides a
special definition for log1p which is more accurate than doing
the naive thing, especially for very small arguments. For example,
the naive version underflows around 2 ** -53, whereas the
specialized version underflows around 2 ** -1074. This function
is used by (+) and (-) on LogFloat.

Definition: expm1 == (subtract 1) . exp. The C language
provides a special definition for expm1 which is more accurate
than doing the naive thing, especially for very small arguments.
This function isn't needed internally, but is provided for
symmetry with log1p.